Rationalise the following: { 5 } / { 3 - sqrt(2) }

The aim here is to turn the fraction so that the denominator does not have a surd. 

Given that we know that any surd squared is equal to the number itself, i.e sqrt(2) * sqrt(2) equas 2, or sqrt(x) * sqrt(x) = x we want to use this rule to try to get rid of the { sqrt(2) } in the question above.

Given however that the denominator is { 3 - sqrt(2) }, the only way to get rid of the surd all together is to multiply both the denominator and the numerator by { 3 + sqrt(2) }. What we did here is reverse the sign. The sign ensures that the surds cancel when we expand the bracket out.

Original fraction to be rationlised: { 5 } / { 3 - sqrt(2) }

Rationalising: { (5) ( 3 + sqrt(2) ) } / { (3 - sqrt(2) ) ( 3 + sqrt(2) ) }

When you multiply everything out you end up with:

{ 15 + 5*sqrt(2) } / { 7 }